axiom//cloudlog.graph Graph Utilities

Author: Temporarily Removed  (
Date: 25 April 2018
Version: 0.4.1

1    Introduction

This is a small graph library intended to provide topological sort.

In this library, graphs are represented as maps from edges to sets of target edges. For example, the map {:a #{:b :c} :b #{c:}} represents a graph with three nodes: :a, :b and :c, where :a is connected to both :b and :c, and :b is connected to :c.

2    merge-graph: Merging Two Graphs

merge-graph merges two graphs by adding the edges of the one to the other

(merge-graph {} {:a #{:b}}) => {:a #{:b}}

Disjoint graphs are merged by simply merging the maps.

(merge-graph {:a #{:b}} {:c #{:d}}) => {:a #{:b} :c #{:d}}

When the two graphs share nodes, their destination sets are merged.

(merge-graph {:a #{:b :c}} {:a #{:c :d}}) => {:a #{:b :c :d}}

3    invert: Invert the Edges of a DGraph

The inverse of an empty graph is an empty graph.

(invert {}) => {}

Each node in the destination set becomes a source edge.

(invert {:a #{:b :c}}) => {:b #{:a} :c #{:a}}

When several edges point at the same node, they are placed in the destination set for that node.

(invert {:a #{:c} :b #{:c}}) => {:c #{:a :b}}

4    souces

sources returns the sources in a graph, i.e., the nodes that do not have other nodes pointing to them.

(sources {:a #{:b :c} :b #{:c}}) => #{:a}

5    toposort: Topological Sort

toposort takes a graph and returns a sequence of its nodes that satisfies the partial order defined by the graph.

(toposort {:a #{:b :c} :b #{:c :d} :c #{:e} :d #{:e}}) => [:a :b :c :d :e]

5.1    Under the Hood

Internally, toposort takes the following arguments:

  1. graph: the graph to be sorted
  2. inv: its inverse
  3. sources: a set of its sources

When sources is empty, it returns an empty sequence.

(toposort {} {} #{}) => empty?

When sources has one element and the graph is empty, the element is returned in the sequence.

(toposort {} {} #{:a}) => [:a]

When the first source has destinations in the graph, we check if they become sources themselves.

(toposort {:a #{:b}} {:b #{:a}} #{:a}) => [:a :b]

However, when the destinations still have other sources (not yet removed from the graph), we do not add them to the sources list.

(let [graph {:a #{:b :c} :b #{:c}}]
 (toposort graph (invert graph) #{:a})) => [:a :b :c]